Volume 2022, Issue 1 9015775

Research Article

Open Access

General Decay of the Moore–Gibson–Thompson Equation with Viscoelastic Memory of Type II

Salah Boulaaras,

Salah Boulaaras

orcid.org/0000-0003-1308-2159

Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia qu.edu.sa

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Abdelbaki Choucha,

Abdelbaki Choucha

Laboratory of Operator Theory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, El Oued, Algeria univ-eloued.dz

Department of Matter Sciences, College of Sciences, Amar Teleji Laghouat University, Laghouat, Algeria

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Andrea Scapellato,

Corresponding Author

Andrea Scapellato

[email protected]

orcid.org/0000-0002-7271-9546

Department of Mathematics and Computer Science, University of Catania, Catania, Italy unict.it

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Salah Boulaaras,

Salah Boulaaras

orcid.org/0000-0003-1308-2159

Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia qu.edu.sa

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Abdelbaki Choucha,

Abdelbaki Choucha

Laboratory of Operator Theory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, El Oued, Algeria univ-eloued.dz

Department of Matter Sciences, College of Sciences, Amar Teleji Laghouat University, Laghouat, Algeria

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Andrea Scapellato,

Corresponding Author

Andrea Scapellato

[email protected]

orcid.org/0000-0002-7271-9546

Department of Mathematics and Computer Science, University of Catania, Catania, Italy unict.it

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First published: 11 March 2022

https://doi.org/10.1155/2022/9015775

Citations: 9

Academic Editor: Gennaro Infante

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Abstract

This study deals with the general decay of solutions of a new class of Moore–Gibson–Thompson equation with respect to the memory kernel of type II. By using the energy method in the Fourier space, we establish the main results.

1. Introduction and Preliminaries

In this work, we are interested in the study of the general decay of solutions of the following problem:

(1)

where a, b, β > 0 are physical parameters and γ is the speed of sound. The convolution term

reflects the memory type-II effect of materials due to viscoelasticity, where h is the relaxation function.

Moore–Gibson–Thompson (MGT) equation is based on the modeling of high amplitude sound waves. There has been quite a bit of work in this area of research due to a wide range of applications such as medical and industrial use of high intensity ultrasound in lithotripsy, heat therapy, ultrasonic cleaning, etc. Classical models of nonlinear acoustics include the Kuznetsov equation, the Westervelt equation, and the Kokhlov–Zabulotskaya–Kuznetsov equation. An in-depth study of linear models is a good starting point for a better understanding of the approximate behaviors of nonlinear models. Indeed, even in the linear case, the works [1] have shown a rich dynamic. In [2], Marchand et al. presented a detailed analysis of this equation. Using a quasi-abstract group approach and refined spectral analysis, they establish the well posedness of the problem and define the accumulation point of eigenvalues. Kaltenbacher et al. [3] also studied the fully nonlinear version of the MGT equation and established the global well posedness and the exponential decay for the nonlinear equation under consideration.

In the case of a third-order equation, the modeling of memory effects is quite complex. The memory term may affect u, or u_t, or even a combination of both. Accordingly, the corresponding models are called as memory of type I, type II, and type III. This classification raises a fundamental question on the impact of the memory terms on the stability properties of the corresponding evolutions. We mention that the memory has some stabilizing effects (see, for instance, [4, 5]). On the contrary, a quantification of the stability results via decay rates shows that the memory may destroy the stability properties of a stable system.

In this paper, we are interested in studying the dynamics that results from the memory kernel of type II (see [6]) in our problem (1).

Natural questions can be asked based on the study of the viscoelastic memory and integral condition system (see [7–10]): could the addition of the memory kernel of type II harm the stability of this kind of problem in any way? If the answer to the question in the wave condition with friction damping are relatively simple, it is not easy to answer in the case of memory kernel of type II that we present below, especially in Fourier space. This work is part of the effort to understand this the MGT problem and memory kernel of type II.

The coupling between the Fourier law of heat condition and different systems has been considered by many authors and there are many results. For example, Bresse system (Bresse–Fourier) has been studied in [11], the Bresse system coupled with the Cattaneo law of heat condition (Bresse–Cattaneo) has been studied in [12], and Timoshenko system with past history has been considered in [13]. For further details, we refer the reader to the following papers [14, 15].

We mention also that several results related to viscoelasticity have been obtained by using the theory of Lie symmetries. Precisely, thanks to the symmetry reduction obtained by means of the classical Lie symmetries, it was possible to obtain exact solutions of considerable interest. For further details in this direction, we refer the reader to [16].

Based on all last mentioned works, especially [6, 10], we would like to prove the general decay result in the Fourier space to problem (1). To the best of our knowledge, this is one of the first works that deals with the MGT problem with viscoelastic memory kernel of type II in the Fourier space.

The paper has the following structure. In Section 1, we put our assumptions and preliminaries that will be employed in our main decay result. In Section 2, by using the energy method in the Fourier space, we construct the Lyapunov functional and find the estimate for the Fourier image. Section 3 is devoted to the conclusion.

To prove our main result, we need the following hypotheses and lemmas.

(H1) h ∈ C¹(ℝ₊, ℝ₊)∩C(ℝ₊, ℝ₊) is a nonincreasing function; there exists a positive constant κ satisfying

(2)

(H2) There exist positive numbers θ and λ ∈ (γ²/b, β/a) such that λ/θ < κ and

(3)

where

and

(H3)

(4)

Throughout this paper, we use c, C, C_i, i = 1,2, to denote a generic positive constant.

Lemma 1. Suppose that (3) holds. Then, for all , we have

(5)

and

(6)

where

(7)

Lemma 2. For any k ≥ 0 and c > 0, there exist a constant C > 0 such that, for all t ≥ 0, the following estimate holds:

(8)

2. The Energy Method in the Fourier Space

In this section, we get the decay estimate of the Fourier image of the solution for system (1). By using Plancherel’s theorem together with some integral estimates such as (2), this method will allow us to give the decay rate of the solution in the energy space. To this goal, we use the energy method in the Fourier space and construct appropriate Lyapunov functionals. Finally, we prove our main result.

Applying the Fourier transform to (1), we get the following problem:

(9)

Lemma 3. Suppose that (3)–(4) hold. Let be the solution of (9). Then, the energy functional , given by

(10)

satisfies

(11)

where C₁ = ((b − γ²/λ) − H(t)(1 + θ/2)) > 0, C₂ = κ₀/2κ, and κ₀ = (κ − λ/θ) > 0.

Proof. Firstly, multiplying (9) by and taking the real part, we obtain

(12)

Then, we have

(13)

and

(14)

Substituting (13) and (14) into (12), we obtain

(15)

Secondly, multiplying (9) by and taking the real part, we obtain

(16)

We have

(17)

and

(18)

Substituting (17) and (18) into (16), we obtain

(19)

At this point, by (4), we have β − γ²a/b > 0, and by (3), we select λ such that γ²/b < λ < β/a.

Let be the energy functional. Then, we have

(20)

By (15) and (19), we have

(21)

(22)

By (3), we have (β − aλ > 0), and using Young’s inequality, we obtain

(23)

By using the fact that λ/θ < κ and under supposition (2), we have

(24)

Thus, (23) becomes

(25)

where κ₀ = (κ − λ/θ) > 0,

Finally, by setting C₁ = (b − γ²/λ − H(t)(1 + θ/2)) > 0 and C₂ = κ₀/2κ, we find (11).

Now, to achieve our goal, we need the following lemmas.

Lemma 4. Suppose that (3)-(4) hold. Then, the functional

(26)

satisfies

(27)

Proof. Differentiating D₁ and by using (9), we obtain

(28)

In what follows, we estimate the terms I_i, i = 1, …, 4, that appear in the right-hand side of (28); using Young’s inequality, we obtain

(29)

(30)

(31)

By letting ε₁ = γ²/6λ, we obtain (27).

Lemma 5. The function,

(32)

satisfies, for any ε, ε₂ > 0,

(33)

Proof. Differentiating D₂ and by using (9), we obtain

(34)

Similarly, we estimate the terms J_i, i = 1, …, 6, that appear in the right-hand side of (34). Using Young’s inequality, we obtain

(35)

(36)

and

(37)

Similarly, we have

(38)

and

(39)

By substituting (35)–(39) into (34), we obtain (33).

At this stage, we define the functional

(40)

where N, N₁, and N₂ are positive constants to be properly chosen later.

Lemma 6. There exist μ_i, t₀ > 0, i = 1, …, 4, such that the functional given by (40) satisfies

(41)

and

(42)

Proof. Since the function h is a positive and continuous, for all t₀ > 0, we have

(43)

Firstly, by differentiating (40) and using (11), (27), and (33), we have

(44)

By setting

(45)

we obtain

(46)

Next, we fixed N₁ and chose N₂ large enough such that

(47)

Hence, we arrive at

(48)

Secondly, we have

(49)

Using Young’s inequality and Lemma 1, we find

(50)

Hence, we obtain

(51)

Now, we choose N large enough such that

(52)

and exploiting (10), estimates (48) and (51), respectively, give

(53)

and

(54)

for some μ_i > 0, i = 1, …, 4.

Theorem 1. Suppose (3)-(4) hold. Then, there exist positive constants d₁ and d₂ such that the energy functional given by (10) satisfies

(55)

where ρ(ξ) = |ξ|²/(1 + |ξ|²).

Proof. From (53) and (2), we have

(56)

Hence,

(57)

By exploiting (54), it can easily be shown that

(58)

Consequently, for some d₂ > 0, we find

(59)

By integrating of (59) over (t₀, t), we obtain

(60)

Hence, by invoking (54), (58), and (60), the continuity, and the boundedness of E, we establish (55).

Now, we state and prove the following result.

Theorem 2. Let k be a nonnegative integer, and suppose that is bounded. Then, satisfies; for all t ≥ 0, the decay estimate is

(61)

Proof. From (10), let

(62)

and we have

and

; then, by applying the Plancherel theorem and exploiting (55), we find

(63)

Now, we estimate M₁, M₂, the low-frequency part |ξ|² ≤ 1, and the high-frequency part |ξ|² ≥ 1.

Firstly, we have ρ(ξ) ≥ 1/2|ξ|², for |ξ|² ≤ 1. Then,

(64)

and, by using Lemma 2, we obtain

(65)

Secondly, we have ρ(ξ) ≥ 1/2, for |ξ|² ≥ 1. Then,

(66)

Substituting (65) and (66) into (63), we find (61).

3. Conclusion

The aim of this work is the study of the general decay estimate of solutions of a new class of Moore–Gibson–Thompson (MGT) equation with respect to the memory kernel of type II by using the energy method in Fourier space. MGT equation is a nonlinear partial differential equation that arises in hydrodynamics and some physical applications. In this paper, we examined the different mechanism resulting from the memory kernel of type II (see [6]), which dictates the emergence of the memory term in the system in the framework of Fourier space.

In the next work, we will try to use the same method with MGT equations, but in light of a generalization of assumption (3). Precisely, we will assume that κ is a positive number and ϑ is a function that fulfills the following conditions:

(67)

Also, we will study the same problem, but with the memory of the type III. In the future, we will try to use the same method with Boussinesq and Hall-MHD equations which are nonlinear partial differential equations that arises in hydrodynamics and some physical applications (see, for example, [17–19]).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally in the writing and editing of this article. All authors read and approved the final version of the manuscript.

Acknowledgments

Andrea Scapellato would like to acknowledge that this work was partially supported by “Piano di incentivi per la ricerca di Ateneo 2020/2022 (Pia.ce.ri),” Università degli Studi di Catania. Andrea Scapellato is a member of the INdAM (Istituto Nazionale di Alta Matematica “Francesco Severi”) Research group GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni).

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Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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General Decay of the Moore–Gibson–Thompson Equation with Viscoelastic Memory of Type II

Abstract

1. Introduction and Preliminaries

2. The Energy Method in the Fourier Space

3. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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General Decay of the Moore–Gibson–Thompson Equation with Viscoelastic Memory of Type II

Abstract

1. Introduction and Preliminaries

2. The Energy Method in the Fourier Space

3. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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